Since original Fuglede's conjecture falses for \(\mathbb {R}^n\) , \(n\ge 3\) , then researchers considered this problem from two persepctives. One is under additional assumptions. In 2003, Iosevich, Katz and Tao [1]} showed that Fuglede's conjecture holds for convex sets in \(\mathbb {R}^{2}\) . Later, a similar result in dimension 3 was proved by Greenfeld and Lev [2]}. Recently, Lev and Matolcsi [3]} proved that Fuglede's conjecture holds for convex domains in \(\mathbb {R}^{n}\)  for all \(n\) . Another is trying to find for which group \(G\) , Fuglede's conjecture holds in \(G\) . In [4]}, [5]}, Fan et al. proved that Fuglede's conjecture holds in \(\mathbb {Q}_{p}\) , the field of \(p\) -adic numbers. We also know that Fuglede's conjecture holds in the following finite Abelian groups: \(\mathbb {Z}_{p}^{d}\)  (\(p=2\)  and \(d\le 6\) ; \(p\)  is an odd prime and \(d=2\) ; \(p=3,5,7\)  and \(d=3\) ) [6]}, [7]}, [8]}, [9]}, \(\mathbb {Z}_{p}\times \mathbb {Z}_{p^{n}}\)  [9]}, [11]}, [12]}, \(\mathbb {Z}_{p}\times \mathbb {Z}_{pq}\)  [13]} and \(\mathbb {Z}_{pq}\times \mathbb {Z}_{pq}\)  [14]}, \(\mathbb {Z}_{p^{n}}\)  [15]}, \(\mathbb {Z}_{p^{n}q^{m}}\)  (\(p<q\)  and \(m\le 9\)  or \(n\le 6\) ; \(p^{m-2}<q^{4}\) ) [16]}, [17]}, [18]}, \(\mathbb {Z}_{pqr}\)  [19]}, \(\mathbb {Z}_{p^{2}qr}\)  [20]} and \(\mathbb {Z}_{pqrs}\)  [21]}, where \(p,q,r,s\)  are distinct primes.
